The above results in a distribution where each pair of \(x_i,x_j\)  variables are \(\rho ^2\) -correlated. Now, the crucial observation here is that, as is the case with noise stability type arguments, the soundness analysis for the above test distribution just relies on the second moment structure, and in particular, just uses the fact that the \((x_i,x_j)\)  variables are pairwise \(\rho ^2\) -correlated. This is due to the folklore observation that for any distribution on \(x_1,\ldots ,x_r\)  that is pairwise \(\rho ^2\) -correlated, using techniques from [1]} one can show
\(_{x_1,\ldots ,x_r} \left[\prod _{i \in [r]} f(x_i)\right] \approx \prod _{i \in [r]} _{x_i}\big [f(x_i)\big ].\) 
